Given an isometric action of a discrete group Γ on a compact manifold M with boundary and a Γ-invariant elliptic boundary value problem D on M, we consider its twisting by projections over the crossed product algebra C ∞(M) ⋊ Γ. The twisted problem is Fredholm and we compute its index in terms of the equivariant Chern character of the principal symbol of D and a noncommutative Chern character of P. In the special case, when D is the Dirichlet problem for the Euler operator, the index is expressed as a linear combination of the Euler characteristics of the xed point submanifolds of the group action.